Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

A line graph $L(G)$ of $G = (V, E)$ is the graph with vertex set $E$ in which $x, y \in E$ are adjacent as vertices if and only if they are adjacent as edges in $G$. In 1970, Beineke (and Robertson independently) discovered a forbidden induced subgraph characterization of the class of line graphs of simple graphs. Bermond and Meyer in 1973 generalized this characterization to the class of line graphs of multigraphs, denoted $\LL$. One such obstruction of these classes is $K_{1,3}$, the claw. In 2008, Chudnovsky and Seymour fully characterized the set of claw-free graphs.   In this paper, we present constructive characterizations of $\FF$-free graphs for several subsets of the seven forbidden graphs of Bermond and Meyer. We develop algorithms to analyze the search space of all graphs without twins that are also connected, doubly-connected (both $G$ and $\bG$ are connected), or irreducible ($G$ is connected and cannot be constructed in specific ways from two graphs containing simplicial vertices). We produce chain theorems and other tools to reduce the problem's complexity in the sense that many proofs can be reduced to finite computation. We implement our algorithms in SageMath to carry out these finite computations and ultimately give constructive characterizations of classes between $\LL$ and claw-free graphs.

Date

4-7-2026

Committee Chair

Ding, Guoli

LSU Acknowledgement

1

LSU Accessibility Acknowledgment

1

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